A lot of effort has been dedicated to studying the behavior of mass-action networks [1]}. In such networks, the probability that two species react is proportional to their abundances, and therefore
the reaction rates are monomials containing only concentrations of reactant species with exponents
equal to the number of molecules entering the reaction. The above example is of this type, but will no
longer be of this type if \(R_1\)  is changed to \(k_1 x_3\) , for instance.
This constraint leads to algebraic properties exploited in chemical reaction network theory (CRNT), which has been initiated by Horn, Jackson and Feinberg [2]}, [1]}.
In order to cover more general models in this paper, we will not impose the mass action
constraint on the reaction rates.
Therefore, our results apply to all systems of polynomial ODEs.
